The space writhes and signatures of polymer knots

TL;DR

This study uses simulations to explore the relationship between space writhe and knot invariants, revealing a strong correlation with the signature for complex knots, thus enhancing understanding of knot topology.

Contribution

It demonstrates that the average space writhe of knotted polymers correlates with the knot signature, extending this relationship to knots with up to 38 crossings.

Findings

Average space writhe matches tight configurations for knots up to 10 crossings.

Strong correlation between space writhe and knot signature for complex knots.

The connection between signature and space writhe persists at high crossing numbers.

Abstract

The space writhe of a knot is a property of its three-dimensional embedding that contains information about its underlying topology, but the correspondence between space writhe and other topological invariants is not fully understood. We perform Langevin dynamics simulations of knotted semiflexible polymers and measure their ensemble average space writhe. We show that for all knots up to 10 crossings, alternating and non-alternating, the average space writhe is almost equal to that of the tightest known configuration of the same knot, with minor differences. Using this equivalence, we show that for more complex knots with up to 38 crossings, the average space writhe is strongly correlated with the signature of the knot. This establishes that the connection between signature and space writhe holds at larger crossing numbers.